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Find Fit second degree parabola equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}} [ Calculator, Method and examples ]

Solution:
Your problem `->` Fit second degree parabola equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}}


The equation is `y = a + bx + cx^2` and the normal equations are

`sum y = an + b sum x + c sum x^2`

`sum xy = a sum x + b sum x^2 + c sum x^3`

`sum x^2y = a sum x^2 + b sum x^3 + c sum x^4`


`X``y``x = X - 1998``x^2``x^3``x^4``x*y``x^2*y`
199640-24-816-80160
199750-11-11-5050
199862000000
19995811115858
20006024816120240
------------------------
999027001003448508


Substituting these values in the normal equations
`270=5a+0b+10c`

`48=0a+10b+0c`

`508=10a+0b+34c`


Solving these 3 equations using inverse matrix method,
Here `5a+10c=270`
`10b=48`
`10a+34c=508`

Now converting given equations into matrix form
`[[5,0,10],[0,10,0],[10,0,34]] [[a],[b],[c]]=[[270],[48],[508]]`

Now, A = `[[5,0,10],[0,10,0],[10,0,34]]`, X = `[[a],[b],[c]]` and B = `[[270],[48],[508]]`

`:. AX = B`

`:. X = A^-1 B`

`|A|` = 
 `5`  `0`  `10` 
 `0`  `10`  `0` 
 `10`  `0`  `34` 


 =
 `5` × 
 `10`  `0` 
 `0`  `34` 
 `+0` × 
 `0`  `0` 
 `10`  `34` 
 `+10` × 
 `0`  `10` 
 `10`  `0` 


`=5 xx (10 × 34 - 0 × 0) +0 xx (0 × 34 - 0 × 10) +10 xx (0 × 0 - 10 × 10)`

`=5 xx (340 +0) +0 xx (0 +0) +10 xx (0 -100)`

`=5 xx (340) +0 xx (0) +10 xx (-100)`






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